How we might attenuate the anti-democratic effects of gerrymandering while simultaneously moderating the negative outcomes of a two-party system.

Congress has the power to get rid of gerrymandering, at least at the federal level, just by passing a law.

The constitutional authority is very clear, in Article I, Section 4:

"The times, places and manner of holding elections for Senators and Representatives, shall be prescribed in each state by the legislature thereof; but the Congress may at any time by law make or alter such regulations, except as to the places of choosing Senators."

The enforcement clauses of the 14th, 15th, and 19th Amendments may plausibly be read as providing additional authority for regulating Congressional election for the purpose of ensuring universal adult suffrage and addressing Equal Protection problems. But this really isn't necessary, given the broad scope of the Article I authority.

One way to do this would be for Congress to legislate an objective algorithm for drawing House districts (several have been proposed by academic studies and think tanks). Another way would be to legislate a set of judicially-enforceable tests for whether a redistricting map were too gerrymandered.

A better way to do it would be to change the voting system to remove the incentive to gerrymander. Gerrymandering is as powerful as it is because current law requires Representatives to be elected from single-member districts, where winning with 51% of the vote is as good as winning with 99%. If Congress were to change this to requiring multi-member districts in states with more than one House seats, and require states to use a voting method that provides a reasonable approximation of proportional representation.

A few options here:

1. Party-list proportional: Each registered party provides a list of candidates. Voters vote for parties, and each party is assigned a number of seats based on their % of the vote. The top candidates from each party's list (up to that party's number of seats) are the winners of the election.
2. Single transferable vote: Voters rank the candidates in preference order, and an Instant Runoff-like system is used to pick the top N candidates to represent the district.
3. Borda Count: Voters rank candidates in preference order, and votes are tallied based on preference rank. E.g. with 10 candidates for a 5-member district, your first choice is tallied as 9 votes, your second as 8 votes, and so on, and the candidates with the 5 highest totals are elected.
4. Limited voting: Voters vote for N candidates, where N is less than the total number of seats. This is tallied like the multimember plurality ballots many California jurisdictions currently use for local elections. For example, you might vote for your three favorite candidates, and then the five highest candidates from the district get elected.

Any of these systems would be much harder and less fruitful to gerrymander than the current system, and all of them would have the further advantage of providing better representation of viewpoints and interests that are substantial minorities of their districts.

Another advantage of this would be to break the perverse effect of the current system of forcing candidates to adhere to one of the two major political parties in order to be viable. If we had an electoral system that had room for 3-5 viable parties in a given race, then we'd see a lot less purely oppositional politics (i.e. parties defining themselves more by being against the other party than by being for any particular thing) and consequently a more productive political debate and a less dysfunctional Congress. Both because of there's no longer a strictly binary choice, and because (with STV or Borda in particular) a big part of how you get elected is by first-choice supporters of other parties being willing to rank you towards the middle rather than in the bottom.

Changing the voting system to make multiple parties potentially viable would also limit the danger when one or both parties' national organization becomes dominated by a toxic leadership. As we've recently seen, capturing the banner of a major party in a two-party system can lead to a situation where anyone who publicly breaks with the leadership has a hard time winning reelection, because they're seen as "betraying" the party (due to the oppositional mindset), and because without votes of the leadership's supporters they likely either lose the primary or lose the general election. It's far from a complete solution, since (barring a Constitutional Amendment) there's only one President at a time and each Senator is elected in a single-member election, but just providing a path for Representatives to be reelected without being publicly allied with one of the two major parties' leadership would be a good start.